Ask the Wizard #366
"Anonymous" .
To answer that, I first looked at the frequency of each letter in each position, based on the list of allowed Wordle solutions.
Letter Frequency in Wordle
Letter | Pos. 1 | Pos. 2 | Pos. 3 | Pos. 4 | Pos. 5 | Total |
---|---|---|---|---|---|---|
A | 141 | 304 | 307 | 163 | 64 | 979 |
B | 173 | 16 | 57 | 24 | 11 | 281 |
C | 198 | 40 | 56 | 152 | 31 | 477 |
D | 111 | 20 | 75 | 69 | 118 | 393 |
E | 72 | 242 | 177 | 318 | 424 | 1233 |
F | 136 | 8 | 25 | 35 | 26 | 230 |
G | 115 | 12 | 67 | 76 | 41 | 311 |
H | 69 | 144 | 9 | 28 | 139 | 389 |
I | 34 | 202 | 266 | 158 | 11 | 671 |
J | 20 | 2 | 3 | 2 | 0 | 27 |
K | 20 | 10 | 12 | 55 | 113 | 210 |
L | 88 | 201 | 112 | 162 | 156 | 719 |
M | 107 | 38 | 61 | 68 | 42 | 316 |
N | 37 | 87 | 139 | 182 | 130 | 575 |
O | 41 | 279 | 244 | 132 | 58 | 754 |
P | 142 | 61 | 58 | 50 | 56 | 367 |
Q | 23 | 5 | 1 | 0 | 0 | 29 |
R | 105 | 267 | 163 | 152 | 212 | 899 |
S | 366 | 16 | 80 | 171 | 36 | 669 |
T | 149 | 77 | 111 | 139 | 253 | 729 |
U | 33 | 186 | 165 | 82 | 1 | 467 |
V | 43 | 15 | 49 | 46 | 0 | 153 |
W | 83 | 44 | 26 | 25 | 17 | 195 |
X | 0 | 14 | 12 | 3 | 8 | 37 |
Y | 6 | 23 | 29 | 3 | 364 | 425 |
Z | 3 | 2 | 11 | 20 | 4 | 40 |
Then I looked at all the words in the Wordle solution list with five distinct letters and scored them according to the letter frequency table above. I awarded two points for a match in the correct position and one point for a match in an incorrect position. Then I sorted the list, which you see below.
Best Starting Words in Wordle
Rank | Word | Points |
---|---|---|
1 | Stare | 5835 |
2 | Arose | 5781 |
3 | Slate | 5766 |
4 | Raise | 5721 |
5 | Arise | 5720 |
6 | Saner | 5694 |
7 | Snare | 5691 |
8 | Irate | 5682 |
9 | Stale | 5665 |
10 | Crate | 5652 |
11 | Trace | 5616 |
12 | Later | 5592 |
13 | Share | 5562 |
14 | Store | 5547 |
15 | Scare | 5546 |
16 | Alter | 5542 |
17 | Crane | 5541 |
18 | Alert | 5483 |
19 | Teary | 5479 |
20 | Saute | 5475 |
21 | Cater | 5460 |
22 | Spare | 5457 |
23 | Alone | 5452 |
24 | Trade | 5449 |
25 | Snore | 5403 |
26 | Grate | 5403 |
27 | Shale | 5392 |
28 | Least | 5390 |
29 | Stole | 5377 |
30 | Scale | 5376 |
31 | React | 5376 |
32 | Blare | 5368 |
33 | Parse | 5351 |
34 | Glare | 5340 |
35 | Atone | 5338 |
36 | Learn | 5324 |
37 | Early | 5320 |
38 | Leant | 5307 |
39 | Paler | 5285 |
40 | Flare | 5280 |
41 | Aisle | 5280 |
42 | Shore | 5274 |
43 | Steal | 5268 |
44 | Trice | 5267 |
45 | Score | 5258 |
46 | Clear | 5258 |
47 | Crone | 5253 |
48 | Stone | 5253 |
49 | Heart | 5252 |
50 | Loser | 5251 |
51 | Taper | 5248 |
52 | Hater | 5243 |
53 | Relay | 5241 |
54 | Plate | 5240 |
55 | Adore | 5239 |
56 | Sauce | 5236 |
57 | Safer | 5235 |
58 | Alien | 5233 |
59 | Caste | 5232 |
60 | Shear | 5231 |
61 | Baler | 5230 |
62 | Siren | 5226 |
63 | Canoe | 5215 |
64 | Shire | 5213 |
65 | Renal | 5210 |
66 | Layer | 5206 |
67 | Tamer | 5200 |
68 | Large | 5196 |
69 | Pearl | 5196 |
70 | Route | 5194 |
71 | Brace | 5192 |
72 | Slice | 5178 |
73 | Stage | 5171 |
74 | Prose | 5170 |
75 | Spore | 5169 |
76 | Rouse | 5166 |
77 | Grace | 5164 |
78 | Solar | 5152 |
79 | Suite | 5150 |
80 | Roast | 5145 |
81 | Lager | 5130 |
82 | Plane | 5129 |
83 | Cleat | 5129 |
84 | Dealt | 5128 |
85 | Spear | 5126 |
86 | Great | 5126 |
87 | Aider | 5123 |
88 | Trope | 5116 |
89 | Spire | 5108 |
90 | Tread | 5107 |
91 | Slave | 5097 |
92 | Close | 5090 |
93 | Lance | 5090 |
94 | Rinse | 5088 |
95 | Cause | 5087 |
96 | Prone | 5087 |
97 | Drone | 5082 |
98 | Noise | 5079 |
99 | Crest | 5073 |
100 | Sober | 5068 |
So, there you have it, my recommended starting word, which I use, is STARE.
Suppose a casino has a game based on a fair coin flip that pays even money. A player wishes to play one million times at $1 a bet. How much money should he bring to the table to have a 50% chance of not going broke?
Ace2
Let's first answer the question of what is the probability the player will be down more than x units after one million flips, assuming the player has an unlimited bankroll.
Since this is a fair bet, the mean win after a million flips is zero. The variance of each flip is 1, so the variance of one million flips is one million. One standard deviation is thus sqrt(1,000,000) = 1000.
We can find the bankroll required with the Excel function =norm.inv(probability,mean,standard deviation). For example, if we put in =norm.inv(.25,0,1000), we get -674.49. This means if after one million flips, the player has a 25% chance of being down 674 or more. Please keep in mind this is an estimate. To get a true answer, we should use the binomial distribution, which would be very tedious with a million flips.
It could very well happen that if the player took $674 to the table, he might run out of money before the million flips. If he could keep playing on credit, it might happen that he has a recovery and finishes less than $674 down. In fact, once the player is at -674, there is a 50/50 chance he will end up above or below -674 at any given point in the future.
So, if the player can play on credit, there are three possible outcomes.
- Player never falls below -674.
- Player falls below -674 at some point, but recovers and finishes above -674.
- Player falls below -674 at some point, keeps playing and loses even more.
We have established scenario 3 has a probability of 25%.
Scenario 2 must have the same probability as scenario 3, because once the player is down -674, he has a 50/50 chance to finish above or below that point after one million flips.
Scenario 1 is the only other alternative, which must have probability 100%-25%-25% = 50%.
If the probability the player never falls below 674 is 50%, then the alternative of falling below must be 100%-50% = 50%.
So, there is our answer to the original question, $674.
This question is asked and discussed in my forum at Wizard of Vegas.
Gialmere
Mark the cards as follows:
1 @ 0.5
1 @ 1.5
2 @ 2.5
1 @ 3.5
2 @ 4.5
1 @ 5.5
1 @ 6.5
This question is asked and discussed in my forum at Wizard of Vegas.